I'm asked to simplify these expressions by using long division. I've attempted these a few times and I'm very confused. Any help you can provide would be very much appreciated.
1. (72 - 8x^2 + 4x^3 - 36x) / (x - 3)
2. (8b^3 - 6) / (2b - 1)
Thank you.
visit calc101.com and click on the link for long division. You can see all the details and play around with other polynomials.
Okay. Thanks so much. :)
Sure! I can help you with these long division problems. Long division is a method used to divide polynomials, which involves dividing the terms of the dividend (the expression being divided) by the divisor (the expression inside the brackets).
Let's start with the first expression:
1. (72 - 8x^2 + 4x^3 - 36x) / (x - 3)
To use long division, we divide the first term of the dividend, which is 4x^3, by the first term of the divisor, which is x. This gives us 4x^2. Now, we multiply the entire divisor, (x - 3), by this quotient, 4x^2. This gives us 4x^3 - 12x^2.
Next, we subtract this result, 4x^3 - 12x^2, from the original dividend, 72 - 8x^2 + 4x^3 - 36x. This can be written as (72 - 8x^2 + 4x^3 - 36x) - (4x^3 - 12x^2).
Simplifying this expression, we get 72 - 8x^2 + 4x^3 - 36x - 4x^3 + 12x^2, which can be rearranged as 72 - 36x + 4x^3 - 8x^2 + 12x^2.
Now, we repeat the process. We divide the first term of this rearranged expression, which is 4x^3, by the first term of the divisor, x. This gives us 4x^2. Again, we multiply the entire divisor, (x - 3), by this quotient, 4x^2. This gives us 4x^3 - 12x^2.
Next, we subtract this result, 4x^3 - 12x^2, from the rearranged expression, which can be written as (72 - 36x + 4x^3 - 8x^2 + 12x^2) - (4x^3 - 12x^2).
Simplifying this expression, we get 72 - 36x + 4x^3 - 8x^2 + 12x^2 - 4x^3 + 12x^2, which simplifies to 72 - 36x.
Therefore, the simplified expression is 72 - 36x.
Now, let's move on to the second expression:
2. (8b^3 - 6) / (2b - 1)
We will follow the same steps as above. Divide the first term of the dividend, which is 8b^3, by the first term of the divisor, 2b. This gives us 4b^2. Multiply the entire divisor, (2b - 1), by this quotient, 4b^2. This gives us 8b^3 - 4b^2.
Next, subtract this result, 8b^3 - 4b^2, from the original dividend, 8b^3 - 6, which can be written as (8b^3 - 6) - (8b^3 - 4b^2).
Simplifying this expression, we get 8b^3 - 6 - 8b^3 + 4b^2, which can be rearranged as 8b^3 - 8b^3 + 4b^2 - 6.
Now, we repeat the process. Divide the first term of this rearranged expression, which is 4b^2, by the first term of the divisor, 2b. This gives us 2b. Multiply the entire divisor, (2b - 1), by this quotient, 2b. This gives us 4b^2 - 2b.
Next, subtract this result, 4b^2 - 2b, from the rearranged expression, which can be written as (8b^3 - 8b^3 + 4b^2 - 6) - (4b^2 - 2b).
Simplifying this expression, we get 8b^3 - 8b^3 + 4b^2 - 6 - 4b^2 + 2b, which further simplifies to 2b - 6.
Therefore, the simplified expression is 2b - 6.
I hope this explanation helps! Let me know if you have any further questions.